
The family of generative temporal models give by Equation REF  include hidden Markov models, non-linear state space models [1]}, and the Variational Recurrent Neural Network (VRNN) [2]}. If we are not modeling temporal or sequential data, then we can assume that the N multivariate observations are indeed exchangeable and obtain the following decomposition instead.
\(p(\mathbf {y}_1,\cdots , \mathbf {y}_N, \mathbf {z}_1,\cdots , \mathbf {z}_N, \theta ) = p(\theta )\prod _{n=1}^N\ p(\mathbf {y}_n \mid \mathbf {z}_n, \theta )p(\mathbf {z}_n\mid \theta )\) 
